OFFSET
1,2
COMMENTS
The old entry with this A-number was a duplicate of A030298.
a(52) is currently unknown. - Donovan Johnson
a(52)-a(10000) were found using a conjunction of Mathematica and Kim Walisch's primecount program. The additional values of the prime-counting function can be found in the second a-file. - Matthew House, Dec 23 2016
LINKS
Matthew House, Table of n, a(n) for n = 1..10000
Farideh Firoozbakht, Notes on the missing terms in this sequence
Matthew House, Values found using primecount
EXAMPLE
1 -> 1; 1 step to see a repeat, so a(1) = 1.
2 -> 1 -> 1; 2 steps to see a repeat.
3 -> 2 -> 1 -> 1; 3 steps to see a repeat.
4 -> 11 -> 5 -> 3 -> 2 -> 1 -> 1; 6 steps to see a repeat.
6 -> 12 -> 112 -> 11114 -> 1733 -> 270 -> 12223 -> 7128 -> 11122225 -> 33991010 -> 13913661 -> 2107998 -> 12222775 -> 33910130 -> 131212367 -> 56113213 -> 6837229 -> 4201627 -> 266366 -> 112430 -> 131359 -> 7981 -> 969 -> 278 -> 134 -> 119 -> 47 -> 15 -> 23 -> 9 -> 22 -> 15; 31 steps to see a repeat.
9 -> 22 -> 15 -> 23 -> 9; 4 steps to see a repeat.
From David Applegate and N. J. A. Sloane, Feb 09 2009: (Start)
The trajectories of the numbers 1 through 17, up to and including the first repeat, are as follows. Note that a(n) is one less than the number of terms shown.
[1, 1]
[2, 1, 1]
[3, 2, 1, 1]
[4, 11, 5, 3, 2, 1, 1]
[5, 3, 2, 1, 1]
[6, 12, 112, 11114, 1733, 270, 12223, 7128, 11122225, 33991010, 13913661, 2107998, 12222775, 33910130, 131212367, 56113213, 6837229, 4201627, 266366, 112430, 131359, 7981, 969, 278, 134, 119, 47, 15, 23, 9, 22, 15]
[7, 4, 11, 5, 3, 2, 1, 1]
[8, 111, 212, 1116, 112211, 52626, 124441, 28192, 11111152, 111165448, 1117261018, 1910112963, 252163429, 42205629, 2914219, 454002, 127605, 231542, 110938, 15631, 44510, 13605, 23155, 3582, 12246, 12637, 1509, 296, 11112, 111290, 131172, 1127117, 76613, 9470, 13161, 21328, 11111114, 14142115, 3625334, 1125035, 348169, 78151, 11369, 1373, 220, 1135, 349, 70, 134, 119, 47, 15, 23, 9, 22, 15]
[9, 22, 15, 23, 9]
[10, 13, 6, 12, 112, 11114, 1733, 270, 12223, 7128, 11122225, 33991010, 13913661, 2107998, 12222775, 33910130, 131212367, 56113213, 6837229, 4201627, 266366, 112430, 131359, 7981, 969, 278, 134, 119, 47, 15, 23, 9, 22, 15]
[11, 5, 3, 2, 1, 1]
[12, 112, 11114, 1733, 270, 12223, 7128, 11122225, 33991010, 13913661, 2107998, 12222775, 33910130, 131212367, 56113213, 6837229, 4201627, 266366, 112430, 131359, 7981, 969, 278, 134, 119, 47, 15, 23, 9, 22, 15]
[13, 6, 12, 112, 11114, 1733, 270, 12223, 7128, 11122225, 33991010, 13913661, 2107998, 12222775, 33910130, 131212367, 56113213, 6837229, 4201627, 266366, 112430, 131359, 7981, 969, 278, 134, 119, 47, 15, 23, 9, 22, 15]
[14, 14]
[15, 23, 9, 22, 15]
[16, 1111, 526, 156, 1126, 1103, 185, 312, 11126, 1734, 1277, 206, 127, 31, 11, 5, 3, 2, 1, 1]
[17, 7, 4, 11, 5, 3, 2, 1, 1]
For n = 18 see A077960.
(End)
MAPLE
with(numtheory):
f := proc(n) local t1, v, r, x, j;
if (n = 1) then return 1; end if;
t1 := ifactors(n): v := 0;
for x in op(2, t1) do r := pi(x[1]):
for j from 1 to x[2] do
v := v * 10^length(r) + r;
end do; end do; v; end proc;
t := proc(n) local v, l, s; v := n; s := {v}; l := [v]; v := f(v);
while not v in s do s := s union {v}; l := [op(l), v]; v := f(v); end do;
[op(l), v];
end proc; [seq(nops(t(n))-1, n=1..17)];
# David Applegate and N. J. A. Sloane, Feb 09 2009
MATHEMATICA
f[n_] := If[n==1, 1, FromDigits@ Flatten[ IntegerDigits@# & /@ (PrimePi@#
& /@ Flatten[ Table[ First@#, {Last@#}] & /@ FactorInteger@n])]];
g[n_] := Length@ NestWhileList[f, n, UnsameQ, All] - 1; Array[g, 39]
(* Robert G. Wilson v, Feb 02 2009; modified slightly by Farideh Firoozbakht, Feb 10 2009 *)
PROG
(GBnums)
void ea (n)
{
mpz u[] ; // factors
mpz tr[]; // sequence
print(n);
while(n > 1)
{
lfactors(u, n); // factorize into u
vmap(u, pi); // replace factors by rank
n = catv(u); // concatenate
print(n);
if(vsearch(tr, n) > 0) break; // loop found
vpush(tr, n); // remember n
}
println('');
}
(Haskell)
import Data.List (genericIndex)
a098282 n = f [n] where
f xs = if y `elem` xs then length xs else f (y:xs) where
y = genericIndex (map a087712 [1..]) (head xs - 1)
-- Reinhard Zumkeller, Jul 14 2013
CROSSREFS
KEYWORD
nonn,base,nice
AUTHOR
Eric Angelini, Feb 02 2009
EXTENSIONS
a(8) and a(10) found by Jacques Tramu
Extended through a(39) by Robert G. Wilson v, Feb 02 2009
Terms through a(39) corrected by Farideh Firoozbakht, Feb 10 2009
a(40)-a(51) from Donovan Johnson, Jan 08 2011
More terms from and a(40) corrected by Matthew House, Dec 23 2016
STATUS
approved